Consider the minimisation and maximisation of the function $f: \\mathbb{R}^2 \\longrightarrow \\mathbb{R}$ defined by $f (x,y) = y + \\simplify{({a}/{b}) * x}$, on the region $D_1 = \\Big\\{ \\begin{pmatrix} x \\\\ y \\end{pmatrix} \\in \\mathbb{R}^2 \\Big| G(x,y) \\leq \\simplify{{r}^2} \\Big\\}$, where $G(x,y) = (x - \\var{centre_x})^2 + (y - \\var{centre_y})^2$.
\nYou should sketch the region $D_1$ and draw some contours of $f$ to establish the existence of a minimum point and a maximum point on the boundary of $D_1$. Consider the direction of increasing contour values. This will help you distinguish between the maximum point and the minimum point.
\nWrite an explicit Lagrangian function for this optimisation point. Following the convention in the lecture notes, please ensure your Langrangian is of the form $L(x,y, \\lambda) = f(x,y) + \\lambda (b - G(x,y))$ for a suitable value of $b$. Due to the current limitations of this system, it will be difficult for you to write $\\lambda$ in the box below, so please use the letter $l$ instead; in the rest of the question we will continue to use $\\lambda$ though).
\n$L(x,y, \\lambda) =$ [[0]]
\nNow find the $x,y$ coordinates, and the associated $\\lambda$, of the minimum point, and find the minimum value. Do the same for the maximum point. (If you require a hint then press the \"Show steps\" button below). For the coordinates, please ensure that all entries are written as fractions; e.g. if you wish to write $\\begin{pmatrix} 1 + \\frac{1}{2} \\\\ 1 + \\frac{1}{4} \\end{pmatrix}$ then write $\\begin{pmatrix} \\frac{3}{2} \\\\ \\frac{5}{4} \\end{pmatrix}$.
\nThe minimum value occurs at $\\begin{pmatrix} x_1 \\\\ y_1 \\end{pmatrix} =$ [[1]] .
The minimum value is $f(x_1 , y_1 ) =$ [[2]] .
The associated value of $\\lambda$ for the minimum point is $\\lambda_1 =$ [[3]] .
The maximum value occurs at $\\begin{pmatrix} x_1 \\\\ y_1 \\end{pmatrix} =$ [[4]] .
The maximum value is $f(x_1 , y_1 ) =$ [[5]] .
The associated value of $\\lambda$ for the maximum point is $\\lambda_1 =$ [[6]] .
The coordinates of the extremum points, and the associated $\\lambda$, will satisfy $L_x (x,y, \\lambda) = 0$ , $L_y (x,y, \\lambda) = 0$ , $L_{\\lambda} (x,y, \\lambda) = 0$.
\nOnce you have obtained these equations, rearrange the first two equations so that you have only $\\lambda$ on the left-hand-sides, and then equate them so you are left with a single expression involving just $x$ and $y$. Your third equation is also an expression involving just $x$ and $y$. You now have two equations with the two unknowns $x$ and $y$, which you can solve to find $x$ and $y$.
\nFinally, you can then use your original equation $L_x (x,y, \\lambda) = 0$ to find the corresponding $\\lambda$.
\nNote that, because we have two extremum points (a maximum and a minimum), you will find two sets of solutions $(x,y,\\lambda)$. In order to determine which one corresponds to the minimum point, and which one corresponds to the maximum point, you should look at the sketch you drew earlier.
", "customMarkingAlgorithm": "", "scripts": {}, "type": "information", "marks": 0, "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "unitTests": []}], "showCorrectAnswer": true, "scripts": {}, "gaps": [{"extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "checkingType": "absdiff", "answer": "y + {a}*x/{b} + l*({r}^2 - (x - {centre_x})^2 - (y - {centre_y})^2)", "expectedVariableNames": [], "checkVariableNames": false, "marks": 1, "showFeedbackIcon": true, "checkingAccuracy": 0.001, "showCorrectAnswer": true, "vsetRange": [0, 1], "scripts": {}, "vsetRangePoints": 5, "showPreview": true, "failureRate": 1, "unitTests": [], "customMarkingAlgorithm": "", "type": "jme"}, {"extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "correctAnswerFractions": true, "numColumns": 1, "markPerCell": false, "marks": "2", "allowResize": false, "showFeedbackIcon": true, "showCorrectAnswer": true, "allowFractions": true, "scripts": {}, "numRows": "2", "correctAnswer": "min_coord", "tolerance": 0, "type": "matrix", "customMarkingAlgorithm": "", "unitTests": []}, {"extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "checkingType": "absdiff", "answer": "{min_value}", "expectedVariableNames": [], "checkVariableNames": false, "marks": 1, "showFeedbackIcon": true, "checkingAccuracy": 0.001, "showCorrectAnswer": true, "vsetRange": [0, 1], "scripts": {}, "vsetRangePoints": 5, "showPreview": true, "failureRate": 1, "unitTests": [], "customMarkingAlgorithm": "", "type": "jme"}, {"extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "checkingType": "absdiff", "answer": "{min_lambda}", "expectedVariableNames": [], "checkVariableNames": false, "marks": 1, "showFeedbackIcon": true, "checkingAccuracy": 0.001, "showCorrectAnswer": true, "vsetRange": [0, 1], "scripts": {}, "vsetRangePoints": 5, "showPreview": true, "failureRate": 1, "unitTests": [], "customMarkingAlgorithm": "", "type": "jme"}, {"extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "correctAnswerFractions": true, "numColumns": 1, "markPerCell": false, "marks": "2", "allowResize": false, "showFeedbackIcon": true, "showCorrectAnswer": true, "allowFractions": true, "scripts": {}, "numRows": "2", "correctAnswer": "max_coord", "tolerance": 0, "type": "matrix", "customMarkingAlgorithm": "", "unitTests": []}, {"extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "checkingType": "absdiff", "answer": "{max_value}", "expectedVariableNames": [], "checkVariableNames": false, "marks": 1, "showFeedbackIcon": true, "checkingAccuracy": 0.001, "showCorrectAnswer": true, "vsetRange": [0, 1], "scripts": {}, "vsetRangePoints": 5, "showPreview": true, "failureRate": 1, "unitTests": [], "customMarkingAlgorithm": "", "type": "jme"}, {"extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "checkingType": "absdiff", "answer": "{max_lambda}", "expectedVariableNames": [], "checkVariableNames": false, "marks": 1, "showFeedbackIcon": true, "checkingAccuracy": 0.001, "showCorrectAnswer": true, "vsetRange": [0, 1], "scripts": {}, "vsetRangePoints": 5, "showPreview": true, "failureRate": 1, "unitTests": [], "customMarkingAlgorithm": "", "type": "jme"}], "unitTests": [], "customMarkingAlgorithm": "", "type": "gapfill"}, {"extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "prompt": "Consider the minimisation of the function $f$ on the region $D_2 = \\Big\\{ \\begin{pmatrix} x \\\\ y \\end{pmatrix} \\in \\mathbb{R}^2 \\Big| G(x,y) \\leq \\simplify{{r}^2} - 10^{-3} \\Big\\}$, where again $G(x,y) =(x - \\var{centre_x})^2 + (y - \\var{centre_y})^2$.
\nBy using your answers to part a, state an approximate value for $f$ at its minimum on $D_2$. For this part of the question, you may wish to look at section 33.5 of the lectures.
\nThe approximate value for $f$ at its minimum on $D_2$ is [[0]] $+$ [[1]] $\\times 10^{-3}$.
", "sortAnswers": false, "marks": 0, "showFeedbackIcon": true, "showCorrectAnswer": true, "scripts": {}, "gaps": [{"extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "checkingType": "absdiff", "answer": "{min_value}", "expectedVariableNames": [], "checkVariableNames": false, "marks": "0.5", "showFeedbackIcon": true, "checkingAccuracy": 0.001, "showCorrectAnswer": true, "vsetRange": [0, 1], "scripts": {}, "vsetRangePoints": 5, "showPreview": true, "failureRate": 1, "unitTests": [], "customMarkingAlgorithm": "", "type": "jme"}, {"extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "checkingType": "absdiff", "answer": "{max_lambda}", "expectedVariableNames": [], "checkVariableNames": false, "marks": "0.5", "showFeedbackIcon": true, "checkingAccuracy": 0.001, "showCorrectAnswer": true, "vsetRange": [0, 1], "scripts": {}, "vsetRangePoints": 5, "showPreview": true, "failureRate": 1, "unitTests": [], "customMarkingAlgorithm": "", "type": "jme"}], "unitTests": [], "customMarkingAlgorithm": "", "type": "gapfill"}, {"extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "prompt": "For the remainder of the question we will look at difference equations.
\nA deposit $D$ is made to an account at the beginning of year $1$ and at the beginning of each subsequent year. The account is earning interest at a nominal interest rate $R$. The interest is compounded annually and is added to the account at the end of each year.
\nLet $s_n$ represent the amount in the account at the beginning of year $n$ just after the deposit $D$ for year $n$ is made.
\ni) What are the first three values in the sequence $\\{ s_n \\}_{n \\in \\mathbb{N}}$?
\n$s_1 =$ [[0]] ;
$s_2 =$ [[1]] ;
$s_3 =$ [[2]] .
ii) Construct a difference equation satisfied by the sequence $\\{ s_n \\}_{n \\in \\mathbb{N}}$, subject to an appropriate initial condition. You answers to part i may help.
\nThe difference equation is $s_{t+1} = \\Big($ [[3]] $\\Big) \\cdot s_t +$ [[4]] ,
subject to the initial condition $s_1 =$ [[5]] .
i) $s_n$ is equal to which of the following? Your answers to part a)i) may help.
\n[[0]]
\nii) So, we can see that $s_n$ is a partial sum of a geometric sequence. Hence, find a simplified expression for $s_n$ in terms of $D$, $R$, and $n$. You may wish to look at section 34.5 of the lecture notes.
\n$s_n =$ [[1]] .
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", "$D (1+R)^{n} + D (1+R)^{n-1} + \\ldots + D (1+R) + D$.
", "$D (1+R)^{n} + D (1+R)^{n-1} + \\ldots + D (1+R)^2 + D (1+R)$.
", "$D (1+R)^{n-1} + D (1+R)^{n-2} + \\ldots + D (1+R)^2 + D (1+R)$.
"], "displayType": "radiogroup", "displayColumns": "1", "showFeedbackIcon": true, "showCorrectAnswer": false, "minMarks": 0, "scripts": {}, "type": "1_n_2", "shuffleChoices": true, "unitTests": [], "maxMarks": 0, "customMarkingAlgorithm": "", "marks": 0}, {"extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "checkingType": "absdiff", "answer": "(D/R) * ((1 + R)^n - 1)", "expectedVariableNames": [], "checkVariableNames": false, "marks": "2", "showFeedbackIcon": true, "checkingAccuracy": 0.001, "showCorrectAnswer": false, "vsetRange": [0, 1], "scripts": {}, "vsetRangePoints": 5, "showPreview": true, "failureRate": 1, "unitTests": [], "customMarkingAlgorithm": "", "type": "jme"}], "unitTests": [], "customMarkingAlgorithm": "", "type": "gapfill"}, {"extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "prompt": "Suppose we told that $D = \\var{{D}}$ and $R = \\var{{IR}}$. At the beginning of which year will we have enough money to buy a house which costs $\\var{{deposit}}$?
\n[[0]]
", "sortAnswers": false, "marks": 0, "showFeedbackIcon": true, "showCorrectAnswer": true, "scripts": {}, "gaps": [{"extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showCorrectAnswer": true, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "marks": "3", "showFeedbackIcon": true, "correctAnswerFraction": false, "mustBeReducedPC": 0, "minValue": "years", "allowFractions": false, "correctAnswerStyle": "plain", "scripts": {}, "maxValue": "years", "type": "numberentry", "customMarkingAlgorithm": "", "unitTests": []}], "unitTests": [], "customMarkingAlgorithm": "", "type": "gapfill"}], "statement": "Lent Term Week 7 (lectures 33 and 34): In this question you will look at the Lagrange's method and first-order linear difference equations.
\nIf you have not provided an answer to every input gap of a question or part of the question, and you try to submit your answers to the question or part, then you will see the message \"Can not submit answer - check for errors\". In reality your answer has been submitted, but the system is just concerned that you have not submitted an answer to every input gap. For this reason, please ensure that you provide an answer to every input gap in the question or part before submitting. Even if you are unsure of the answer, write down what you think is most likely to be correct; you can always change your answer or retry the question.
\nAs with all questions, there may be parts where you can choose to \"Show steps\". This may give a hint, or it may present sub-parts which will help you to solve that part of the question. Furthermore, remember to always press the \"Show feedback\" button at the end of each part. Sometimes, helpful feedback will be given here, and often it will depend on how correctly you have answered and will link to other parts of the question. Hence, always retry the parts until you obtain full marks, and then look at the feedback again.
Keep in mind that in order to see the feedback for a particular part of a question, you must provide a full (but not necessarily correct) answer to that part. Do not worry though, as you can look at the feedback and then ammend your answer accordingly.
Furthermore, as with all questions, choosing to reveal the answers will only show you the answers which change every time the question is loaded (i.e. answers to randomised questions); the fixed answers will not be revealed.
This is the question for Lent Term week 7 of the MA100 course at the LSE. It looks at material from chapters 33 and 34.
\nThe following is a description of parts a and b. In particular it describes the varaibles used for those parts.
\nThis question (parts a and b) looks at optimisation problems using the langrangian method. parts a and b of the question we will ask the student to optimise the objective function f(x,y) = y + (a/b)x subject to the constraint function r^2 = (x-centre_x)^2 + (y-centre_y)^2.
\nThe variables centre_x and centre_y take values randomly chosen from {6,7,...,10} and r takes values randomly chosen from {1,2,...,5}.
\nWe have the ordered set of variables (a,b,c) defined to be randomly chosen from one of the following pythagorean triplets: (3,4,5) , (5,12,13) , (8,15,17) , (7,24,25) , (20,21,29). The a and b variables here are the same as those in the objective function. They are defined in this way because the minimum will occur at (centre_x - (a/c)*r , centre_y - (b/c)*r) with value centre_y - (b/c)r + (a/b) * centre_x - (a^2/bc)*r , and the maximum will occur at (centre_x + (a/c)*r , centre_y + (b/c)*r) with value centre_y + (b/c)r + (a/b) *centre_x + (a^2/bc)r. The minimisation problem has lambda = -c/(2br) and the maximation problem has lambda* = c/(2br).
\nWe can see that all possible max/min points and values are nice rational numbers, yet we still have good randomisation in this question. :)
"}, "advice": "", "variables": {"IR": {"definition": "random(0.01 , 0.02 , 0.03 , 0.04 , 0.05)", "templateType": "anything", "name": "IR", "group": "Variables for part e", "description": "IR+1 is the common ratio of our geometric sequence. IR stands for interest rate.
"}, "min_lambda": {"definition": "-c/(2*b*r)", "templateType": "anything", "name": "min_lambda", "group": "Variables for parts a and b", "description": "This is the associated lambda of our minimum point that we get when using Langrange's method.
"}, "max_lambda": {"definition": "c/(2*b*r)", "templateType": "anything", "name": "max_lambda", "group": "Variables for parts a and b", "description": "This is the associated lambda of our maximum point that we get when using Langrange's method.
"}, "list1": {"definition": "[3,4,5]", "templateType": "anything", "name": "list1", "group": "Variables for parts a and b", "description": "See the description of this question in the settings tab.
"}, "min_coord": {"definition": "vector(centre_x - a*r/c , centre_y - b*r/c)", "templateType": "anything", "name": "min_coord", "group": "Variables for parts a and b", "description": "These are the coordinates of the minimum point of our optimisation problem.
"}, "centre_x": {"definition": "random(6..10)", "templateType": "anything", "name": "centre_x", "group": "Variables for parts a and b", "description": "See the description of this question, in the settings tab, for a more detailed description of the question and the variables.
This variable is the x coordinate of the centre of our constraint circle.
These are the coordinates of the maximum point of our optimisation problem.
"}, "c": {"definition": "list[2]", "templateType": "anything", "name": "c", "group": "Variables for parts a and b", "description": "See the description of this question in the settings tab.
"}, "list2": {"definition": "[5,12,13]", "templateType": "anything", "name": "list2", "group": "Variables for parts a and b", "description": "See the description of this question in the settings tab.
"}, "min_value": {"definition": "centre_y + a*centre_x/b - c*r/b", "templateType": "anything", "name": "min_value", "group": "Variables for parts a and b", "description": "This is the minimum value of our optimisation problem.
"}, "a": {"definition": "list[0]", "templateType": "anything", "name": "a", "group": "Variables for parts a and b", "description": "See the description of this question in the settings tab.
"}, "years": {"definition": "ceil((log(IR * deposit / D + 1))/(log(1 + IR)))", "templateType": "anything", "name": "years", "group": "Variables for part e", "description": "This is the answer to part e. It is number of years we need until we have enough money for the deposit.
"}, "list": {"definition": "random(list1 , list2 , list3 , list4 , list5)", "templateType": "anything", "name": "list", "group": "Variables for parts a and b", "description": "See the description of this question in the settings tab.
"}, "r": {"definition": "random(1..5)", "templateType": "anything", "name": "r", "group": "Variables for parts a and b", "description": "This is the radius of our constraint circle.
"}, "max_value": {"definition": "centre_y + a*centre_x/b + c*r/b", "templateType": "anything", "name": "max_value", "group": "Variables for parts a and b", "description": "This is the maximum value of our optimisation problem.
"}, "list4": {"definition": "[7,24,25]", "templateType": "anything", "name": "list4", "group": "Variables for parts a and b", "description": "See the description of this question in the settings tab.
"}, "centre_y": {"definition": "random(6..10)", "templateType": "anything", "name": "centre_y", "group": "Variables for parts a and b", "description": "This is the y coordinate of the centre of our constraint circle.
"}, "b": {"definition": "list[1]", "templateType": "anything", "name": "b", "group": "Variables for parts a and b", "description": "See the description of this question in the settings tab.
"}, "deposit": {"definition": "random(100000 , 200000 , 300000 , 400000 , 500000)", "templateType": "anything", "name": "deposit", "group": "Variables for part e", "description": "This is the deposite we need in part e.
"}, "list5": {"definition": "[20,21,29]", "templateType": "anything", "name": "list5", "group": "Variables for parts a and b", "description": "See the description of this question in the settings tab.
"}, "new_min": {"definition": "min_value - min_lambda*(10^(-3))", "templateType": "anything", "name": "new_min", "group": "Variables for parts a and b", "description": "This is the new minimum value, for part b.
"}, "list3": {"definition": "[8,15,17]", "templateType": "anything", "name": "list3", "group": "Variables for parts a and b", "description": "See the description of this question in the settings tab.
"}, "D": {"definition": "random(1000 , 2000 , 3000 , 4000 , 5000)", "templateType": "anything", "name": "D", "group": "Variables for part e", "description": "This is the first term of our geometric sequence.
"}}, "ungrouped_variables": [], "extensions": [], "preamble": {"js": "", "css": ""}, "type": "question", "contributors": [{"name": "Michael Yiasemides", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1440/"}]}]}], "contributors": [{"name": "Michael Yiasemides", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1440/"}]}